Uncertainty Inequalities for the Linear Canonical Hilbert Transform

The Heisenberg uncertainty principle plays an important role in signal processing, applied mathematical and physics community. The extensions of the traditional uncertainty principle to the novel transforms have received considerable attention recently. In this paper, the uncertainty inequalities under the linear canonical Hilbert transform (LCHT) are considered. First, the uncertainty principle for the complex signals associated with the one-dimensional LCHT is investigated. Subsequently, the uncertainty inequality for the complex signals derived by the two-dimensional LCHT is proposed. It is shown that the uncertainty bounds for the complex signals derived by LCHT are different from the general complex signals in the LCT domain. In addition, the example and potential applications are also presented to show the correctness and useful of the derived results. Therefore, the uncertainty principles presented here not only can enrich the ensemble of the classical uncertainty principle, but can have many real applications too.